High-order Total Variation Research Articles

Poisson noise removal based on non-convex hybrid regularizers

The presence of TV regularizer always induces an unsatisfactory staircase effect. To overcome the staircase while better sustaining edge information, this work proposes a novel model for Poisson noise removal. The model is based on non-convex mixed regularizers, which involves introducing a non-convex penalty into a composition of the total variation and the higher-order total variation. The iterative reweighted l1 algorithm was used to convert the non-convex model into a convex one. The classic alternating direction method of multipliers was then employed to obtain approximate solutions of the model. When applying this model to degraded images contaminated by Poisson noise of medium to high intensity, its performance in noise suppression was tested. The regularizer was compared with others in the terms of the visual effect of the picture, time cost, and several commonly accepted quantitative indicators for evaluation, such as peak signal-to-noise ratio, feature similarity index and structural similarity index. Numerical experiments showed that the present model not only eliminates block artifacts but also retains sharp edges.

Cauchy noise removal using high-order total variation and overlapping group sparsity

Image denoising presents a significant challenge in computer vision, aiming to eliminate unwanted noise from images and restore their original quality. Cauchy noise, characterized by its heavy-tailed distribution, poses a unique hurdle among various types of noise. While several denoising models have been proposed, Total Variation (TV)-based methods, such as the Remove Cauchy Noise model, are widely used for their effectiveness. However, these approaches often suffer from issues such as oversmoothing, staircase artifacts, and the introduction of false details. To address these limitations, recent research has explored the combination of high-order TV with Overlapping Group Sparsity (OGS), showing promising results in noise removal. Inspired by this, our article introduces a novel Cauchy denoising model. Our approach leverages OGS and directional higher-order TV to effectively remove Cauchy noise while preserving image details and minimizing aliasing and smoothing artifacts. The Chambolle-Pock algorithm efficiently solves the underlying optimization problem. Through qualitative and quantitative evaluations, including visualization and parameter measurements, we demonstrate the competitiveness of our model compared to existing methods.

Total Third-Degree Variation for Noise Reduction in Atomic-Resolution STEM Images.

Scanning Transmission Electron Microscopy (STEM) enables direct determination of atomic arrangements in materials and devices. However, materials such as battery components are weak for electron beam irradiation and low electron doses are required to prevent beam-induced damages. Noise removal is thus essential for precise structural analysis of electron beam sensitive materials at atomic resolution. Total square variation (TSV) regularization is an algorithm that exhibits high noise removal performance. However, the use of the TSV regularization term leads to significant image blurring and intensity reduction. To address these problems, we here propose a new approach adopting L2 norm regularization based on higher-order total variation. An atomic-resolution STEM image can be approximated as a set of smooth curves represented by quadratic functions. Since the third-degree derivative of any quadratic function is 0, total third-degree variation (TTDV) is suitable for a regularization term. The application of TTDV for denoising the atomic-resolution STEM image of CaF2 observed along the [001] zone axis is shown, where we can clearly see the Ca and F atomic columns without compromising image quality.

Mixed Gaussian-impulse noise removal using non-convex high-order TV penalty

To restore images with clear edge details and no staircase artifacts from degraded versions, this paper incorporates the ℓ2 plus ℓ0 data fidelity and non-convex high-order total variation regularizer to establish an optimization model for eliminating mixed Gaussian-impulse noise. Among them, the ℓ2 fidelity is adopted to suppress Gaussian noise, while the ℓ0-norm is more suitable for detecting and removing impulse noise. In addition, the non-convex regularization displays excellent performance in overcoming the staircase effect and preserving edge details. Computationally, by using the iteratively reweighted ℓ1 algorithm and variable splitting method, this work designs a modified alternating minimization method to solve the optimization problem we construct. In theory, the convergence proof of our resulting algorithm is also presented. Finally, in the experimental section, we conduct extensive numerical experiments on degraded images and compare with other existing techniques. From the intuitive effects and restoration accuracy, it follows that our newly proposed method is effective and competitive for image deblurring and mixed noise removal.

Few-shot segmentation framework for lung nodules via an optimized active contour model.

Accurate segmentation of lung nodules is crucial for the early diagnosis and treatment of lung cancer in clinical practice. However, the similarity between lung nodules and surrounding tissues has made their segmentation a longstanding challenge. Existing deep learning and active contour models each have their limitations. This paper aims to integrate the strengths of both approaches while mitigating their respective shortcomings. In this paper, we propose a few-shot segmentation framework that combines a deep neural network with an active contour model. We introduce heat kernel convolutions and high-order total variation into the active contour model and solve the challenging nonsmooth optimization problem using the alternating direction method of multipliers. Additionally, we use the presegmentation results obtained from training a deep neural network on a small sample set as the initial contours for our optimized active contour model, addressing the difficulty of manually setting the initial contours. We compared our proposed method with state-of-the-art methods for segmentation effectiveness using clinical computed tomography (CT) images acquired from two different hospitals and the publicly available LIDC dataset. The results demonstrate that our proposed method achieved outstanding segmentation performance according to both visual and quantitative indicators. Our approach utilizes the output of few-shot network training as prior information, avoiding the need to select the initial contour in the active contour model. Additionally, it provides mathematical interpretability to the deep learning, reducing its dependency on the quantity of training samples.

A non-local total generalized variation regularization reconstruction method for sparse-view x-ray CT

Sparse-view x-ray computed tomography (CT) reconstruction, employing total generalized variation (TGV), effectively mitigates the stepwise artifacts associated with total variation (TV) regularization while preserving structural features within transitional regions of the reconstructed image. Despite TGV surpassing TV in reconstruction quality, it neglects the non-local self-similarity prior, recognized for its efficacy in restoring details during CT reconstruction. This study introduces a non-local TGV (NLTGV) to address the limitation of TGV regularization method. Specifically, we propose an NLTGV-regularized method for sparse-view CT reconstruction, utilizing non-local high-order derivative information to maintain image features and non-local self-similarity for detail recovery. Owing to the non-differentiability of the NLTGV regularized, we employ an alternating direction method of multipliers optimization method, facilitating an efficient solution by decomposing the reconstruction model into sub-problems. The proposed method undergoes evaluation using both simulated and real-world projection data. Simulation and experimental results demonstrate the efficacy of the proposed approach in enhancing the quality of reconstructed images compared to other competitive variational reconstruction methods. In conclusion, the simultaneous incorporation of sparsity priors of high-order TV and non-local similarity proves advantageous for structural detail recovery in sparse-view CT reconstruction.

An Image-Denoising Framework Using ℓq Norm-Based Higher Order Variation and Fractional Variation with Overlapping Group Sparsity

As one of the most significant issues in imaging science, image denoising plays a major role in plenty of image processing applications. Due to the ill-posed nature of image denoising, total variation regularization is widely used in image denoising problems for its capability to suppress noise and preserve image edges. Nevertheless, traditional total variation will inevitably yield undesirable staircase artifacts when applied to recorded images. Inspired by the success of ℓq norm minimization and overlapping group sparsity in image denoising, and the effective staircase artifacts removal by fractional total variation, the hybrid model which combines the fractional order total variation with overlapping group sparsity and higher order total variation with ℓq norm is developed in this paper to restore images corrupted by Gaussian noise. An efficient algorithm based on the parallel linear alternating direction method of multipliers is developed for solving the corresponding model and the numerical experiments demonstrate the effectiveness of the proposed approach against several state-of-the-art methods, in terms of peak signal-to-noise ratio and structure similarity index measure values.

A Non-Convex Hybrid Overlapping Group Sparsity Model with Hyper-Laplacian Prior for Multiplicative Noise

Multiplicative noise removal is a quite challenging problem in image denoising. In recent years, hyper-Laplacian prior information has been successfully introduced in the image denoising problem and significant denoising effects have been achieved. In this paper, we propose a new hybrid regularizer model for removing multiplicative noise. The proposed model consists of the non-convex higher-order total variation and overlapping group sparsity on a hyper-Laplacian prior regularizer. It combines the advantages of the non-convex regularization and the hybrid regularization, which may simultaneously preserve the fine-edge information and reduce the staircase effect at the same time. We develop an effective alternating minimization method for the proposed nonconvex model via an alternating direction method of multipliers framework, where the majorization–minimization algorithm and the iteratively reweighted algorithm are adopted to solve the corresponding subproblems. Numerical experiments show that the proposed model outperforms the most advanced model in terms of visual quality and certain image quality measurements.

Adaptive-weighted high order TV algorithm for sparse-view CT reconstruction.

With the development of low-dose computed tomography (CT), incomplete data reconstruction has been widely concerned. The total variation (TV) minimization algorithm can accurately reconstruct images from sparse or noisy data. However, the traditional TV algorithm ignores the direction of structures in images, leading to the loss of edge information and block artifacts when the object is not piecewise constant. Since the anisotropic information can facilitate preserving the edge and detail information in images, we aim to improve the TV algorithm in terms of reconstruction accuracy via this approach. In this paper, we propose an adaptive-weighted high order total variation (awHOTV) algorithm. We construct the second order TV-norm using the second order gradient, adapt the anisotropic edge property between neighboring image pixels, adjust the local image-intensity gradient to keep edge information, and design the corresponding Chambolle-Pock (CP) solving algorithm. Implementing the proposed algorithm, comprehensive studies are conducted in the ideal projection data experiment where the Structural similarity (SSIM), Root Mean Square Error (RMSE), Contrast to noise ratio (CNR), and modulation transform function (MTF) curves are utilized to evaluate the quality of reconstructed images in statism, structure, spatial resolution, and contrast, respectively. In the noisy data experiment, we further use the noise power spectrum (NPS) curve to evaluate the reconstructed images and compare it with other three algorithms. We use the 2D slice in the XCAT phantom, 2D slice in TCIA Challenge data and FORBILD phantom as simulation phantoms and use real bird data for real verification. The results show that, compared with the traditional TV and FBP algorithms, the awHOTV has better performance in terms of RMSE, SSIM, and Pearson correlation coefficient (PCC) under the projected data with different sparsity. In addition, the awHOTV algorithm is robust against the noise of different intensities. The proposed awHOTV method can reconstruct the images with high accuracy under sparse or noisy data. The awHOTV solves the strip artifacts caused by sparse data in the FBP method. Compared with the TV method, the awHOTV can effectively suppress block artifacts and has good detail protectionability.

On the role of tensor interpolation in solving high-WI viscoelastic fluid flow

The high Weissenberg number (Wi) problem (HWNP) has long been a challenge of viscoelastic fluid flow simulation. This Letter points out that the tensor interpolation method during solving the differential constitutive equations is the main origin of the loss of symmetric positive-definite (SPD) property of the conformation tensor, which is the trigger of the HWNP. Instead of component-based interpolation, we propose a tensor-based interpolation method for the conformation tensor, which is essentially SPD, and the results show that this method is very effective in dealing with the HWNP by significantly improving the numerical accuracy on the invariants of conformation tensor as well as greatly improving the SPD property of the conformation tensor. Moreover, the high-order total variation diminishing schemes can also be easily constructed and applied to solve high-Wi viscoelastic fluid flow under the proposed framework without adding artificial diffusion.

Estimation and Uncertainty Quantification for Piecewise Smooth Signal Recovery

This paper presents a sparse Bayesian learning (SBL) algorithm for linear inverse problems with a high order total variation (HOTV) sparsity prior. For the problem of sparse signal recovery, SBL often produces more accurate estimates than maximum a posteriori estimates, including those that rely on l1 regularization. Moreover, rather than a single signal estimate, SBL yields a full posterior density estimate which can be used for uncertainty quantification. However, SBL is only immediately applicable to problems having a direct sparsity prior, or to those that can be formed via synthesis. This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis, and then develops a corresponding Bayesian learning method. This expands the class of problems available to Bayesian learning to include, e.g., inverse problems dealing with the recovery of piecewise smooth functions or signals from data. Numerical examples are provided to demonstrate how this new technique is effectively employed.

An adaptive image restoration algorithm based on hybrid total variation regularization

In imaging systems, the mixed Poisson-Gaussian noise (MPGN) model can accurately describe the noise present. Total variation (TV) regularization-based methods have been widely utilized for Poisson-Gaussian removal with edge-preserving. However, TV regularization sometimes causes staircase artifacts with piecewise constants. To overcome this issue, we propose a new model in which the regularization term is represented by a combination of total variation and high-order total variation. We study the existence and uniqueness of the minimizer for the considered model. Numerically, the minimization problem can be efficiently solved by the alternating minimization method. Furthermore, we give rigorous convergence analyses of our algorithm. Experiments results are provided to demonstrate the superiority of our proposed hybrid model and algorithm for deblurring and denoising images simultaneously, in comparison with several state-of-the-art numerical algorithms.

Adaptive weighted curvature-based active contour for ultrasonic and 3T/5T MR image segmentation

Image segmentation is a complex and core technique for disease diagnosis or image-guided surgery in the medical image domain. However, low-quality images, such as images with weak edges and intensity inhomogeneities, may bring considerable challenges for radiologists. In this paper, we propose an adaptive weighted curvature-based active contour model by coupling heat kernel convolution and adaptively weighted high-order total variation for medical image segmentation to improve diagnosis effectiveness. To reduce the computational complexity, the heat kernel convolution operation is applied to approximate the perimeter of a segmentation curve. In addition, the weighted parameter included in the high-order total variation term can be automatically evaluated based on an adaptive input image to emphasize local structures and increase segmentation accuracy. Since the proposed method is a smoothing optimization model, the alternating direction method of multipliers is introduced to translate the original problems into several easily solvable subproblems. The numerical experimental results on ultrasonic and 3T/5T MRI datasets demonstrate that the proposed model is quite efficient and robust compared with several traditional segmentation methods.

Non-Convex High-Order TV and ℓ 0-Norm Wavelet Frame-Based Speckle Noise Reduction

To obtain natural restorations from the noisy images contaminated by speckle noise, this brief presents a novel hybrid non-convex regularizers model for image denoising. The proposed new variational model closely combines the superiorities of non-convex high-order total variation function and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{0}$ </tex-math></inline-formula> -norm wavelet frame. This combination helps to avoid the staircase artifacts and maintain discontinuities while removing noise. Numerically, by integrating two popular tools: iteratively reweighted <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> algorithm and variable splitting method, a modified alternating minimization method is adopted to optimize the resulting minimization problem. Finally, compared with several despeckling methods, numerical experiments indicate the competitive performance of our solver in visual improvement and objective measurement.

Hybrid non-convex regularizers model for removing multiplicative noise

Obtaining natural and realistic restorations from the noisy images contaminated by multiplicative noise is a challenging task in image processing. To get over this conundrum, by introducing the non-convex potential functions into the total variation and high-order total variation regularizers, we investigate a novel hybrid non-convex optimization model for image restoration. Numerically, to optimize the resulting high-order PDE system, a proximal linearized alternating minimization method, based on the classical iteratively reweighted ℓ 1 algorithm and variable splitting technique, is designed in detail. Meanwhile, the convergence of the constructed algorithm is also established on the basis of convex analysis. The provided numerical experiments point out that our new scheme shows superiorities in both visual effects and quantitative comparison, especially in terms of the staircase aspects suppression and edge details preservation, compared with some popular denoising methods.

Numerical simulation of Allen–Cahn equation with nonperiodic boundary conditions by the local discontinuous Galerkin method

In this paper, the local discontinuous Galerkin method is used to analyze numerical solutions for nonlinear Allen–Cahn equations with nonperiodic boundary conditions. To begin with, the spatial variables are discretized to generate a semidiscrete method of lines scheme. This yields an ordinary differential equation system in the temporal variable, which is then solved using the higher-order total variation diminishing Runge–Kutta method. A comparison of the generated numerical results to the exact results for various test problems using different tables and figures provides insight into the effectiveness and accuracy of the proposed method. The numerical results confirm that the proposed method is an effective numerical scheme for solving the Allen–Cahn equation since the obtained solutions are extremely close to the exact solutions while exhibiting substantially less error.

Developing a Reliable Holographic Flow Cyto-Tomography Apparatus by Optimizing the Experimental Layout and Computational Processing.

Digital Holographic Tomography (DHT) has recently been established as a means of retrieving the 3D refractive index mapping of single cells. To make DHT a viable system, it is necessary to develop a reliable and robust holographic apparatus in order that such technology can be utilized outside of specialized optics laboratories and operated in the in-flow modality. In this paper, we propose a quasi-common-path lateral-shearing holographic optical set-up to be used, for the first time, for DHT in a flow-cytometer modality. The proposed solution is able to withstand environmental vibrations that can severely affect the interference process. Furthermore, we have scaled down the system while ensuring that a full 360° rotation of the cells occurs in the field-of-view, in order to retrieve 3D phase-contrast tomograms of single cells flowing along a microfluidic channel. This was achieved by setting the camera sensor at 45° with respect to the microfluidic direction. Additional optimizations were made to the computational elements to ensure the reliable retrieval of 3D refractive index distributions by demonstrating an effective method of tomographic reconstruction, based on high-order total variation. The results were first demonstrated using realistic 3D numerical phantom cells to assess the performance of the proposed high-order total variation method in comparison with the gold-standard algorithm for tomographic reconstructions: namely, filtered back projection. Then, the proposed DHT system and the processing pipeline were experimentally validated for monocytes and mouse embryonic fibroblast NIH-3T3 cells lines. Moreover, the repeatability of these tomographic measurements was also investigated by recording the same cell multiple times and quantifying the ability to provide reliable and comparable tomographic reconstructions, as confirmed by a correlation coefficient greater than 95%. The reported results represent various steps forward in several key aspects of in-flow DHT, thus paving the way for its use in real-world applications.

Analysis of Hypergraph Signals via High-Order Total Variation

Beyond pairwise relationships, interactions among groups of agents do exist in many real-world applications, but they are difficult to capture by conventional graph models. Generalized from graphs, hypergraphs have been introduced to describe such high-order group interactions. Inspired by graph signal processing (GSP) theory, an existing hypergraph signal processing (HGSP) method presented a spectral analysis framework relying on the orthogonal CP decomposition of adjacency tensors. However, such decomposition may not exist even for supersymmetric tensors. In this paper, we propose a high-order total variation (HOTV) form of a hypergraph signal (HGS) as its smoothness measure, which is a hyperedge-wise measure aggregating all signal values in each hyperedge instead of a pairwise one in most existing work. Further, we propose an HGS analysis framework based on the Tucker decomposition of the hypergraph Laplacian induced by the aforementioned HOTV. We construct an orthonormal basis from the HOTV, by which a new spectral transformation of the HGS is introduced. Then, we design hypergraph filters in both vertex and spectral domains correspondingly. Finally, we illustrate the advantages of the proposed framework by applications in label learning.

Remove the salt and pepper noise based on the high order total variation and the nuclear norm regularization

This paper proposes a new model to remove the salt and pepper (SAP) noise problem. In the proposed method, we combine the high order total variation regularization with the nuclear norm regularization in order to keep details and structures of the restored images. Since the proposed model is convex and separable, the classic alternating direction method of multipliers can be employed to solve it by introducing some auxiliary variables to transform the original problem into the saddle point problem. Theoretically, we establish the convergence analysis of the proposed numerical algorithm. Final experimental comparisons are provided to show the satisfactory performance of the proposed model, which outperforms other related competitive methods in both the SNR and the SSIM.

Super Resolution CT Imaging Using Higher Order Total Variation (HOTV) Technique

The present study focuses on the performance upgradation of a table-top computed tomography (CT) scanning instrument. The size restriction of the detectors of table-top CT scanner imposes limitation on the allowable size of the object to be scanned. The object with increasing size has to be put in a far end from the X-ray source to be fit into the scan field of view of the scanner. This situation affects the pixel resolution of the tomography images. The limited resolution capacity restricts the CT image zoomed-in view facility which is very often an essential requirement for scientific measurement and analysis. The hardware-based solution involving larger detectors will significantly increase the monetary cost, and thus a cost-effective solution is very much required. The new super resolution (SR) or resolution up-scaling technique based on higher order total variation minimization has been proposed to overcome the above-mentioned problem associated with a table-top tomography instrument. The proposed technique is capable of producing SR image from a single low-resolution image. The experimental results have been compared with some of the existing SR techniques, in terms of various image quality parameters (mean squared error, correlation coefficient, structural similarity index, and feature similarity index) to show betterment over other existing SR techniques. The experiments are carried out with different objects to validate the universality of the proposed scheme. The pixel-enhanced tomographic images are further checked with Sobolev space error analysis to ensure the quality of the images from tomography perspective. It can be inferred that the presented technique can properly up-scale the resolution of CT images, and therefore it presents a cost-effective possible solution to the resolution problem caused by the size constraint of the detectors of table-top CT scanning instruments.